Properties

Label 2-3e2-1.1-c21-0-5
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $25.1529$
Root an. cond. $5.01527$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.84e3·2-s + 5.99e6·4-s − 3.10e6·5-s + 3.63e8·7-s + 1.10e10·8-s − 8.84e9·10-s − 1.45e10·11-s + 1.13e11·13-s + 1.03e12·14-s + 1.89e13·16-s + 8.58e12·17-s − 2.92e13·19-s − 1.86e13·20-s − 4.14e13·22-s + 1.55e14·23-s − 4.67e14·25-s + 3.22e14·26-s + 2.17e15·28-s − 2.40e15·29-s + 2.23e15·31-s + 3.06e16·32-s + 2.44e16·34-s − 1.12e15·35-s − 3.07e16·37-s − 8.30e16·38-s − 3.44e16·40-s + 1.03e17·41-s + ⋯
L(s)  = 1  + 1.96·2-s + 2.85·4-s − 0.142·5-s + 0.486·7-s + 3.64·8-s − 0.279·10-s − 0.169·11-s + 0.228·13-s + 0.954·14-s + 4.30·16-s + 1.03·17-s − 1.09·19-s − 0.406·20-s − 0.332·22-s + 0.784·23-s − 0.979·25-s + 0.447·26-s + 1.38·28-s − 1.05·29-s + 0.490·31-s + 4.80·32-s + 2.02·34-s − 0.0692·35-s − 1.05·37-s − 2.14·38-s − 0.519·40-s + 1.20·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(25.1529\)
Root analytic conductor: \(5.01527\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(7.109010408\)
\(L(\frac12)\) \(\approx\) \(7.109010408\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 711 p^{2} T + p^{21} T^{2} \)
5 \( 1 + 124398 p^{2} T + p^{21} T^{2} \)
7 \( 1 - 51900560 p T + p^{21} T^{2} \)
11 \( 1 + 1325621196 p T + p^{21} T^{2} \)
13 \( 1 - 113350790702 T + p^{21} T^{2} \)
17 \( 1 - 505258211646 p T + p^{21} T^{2} \)
19 \( 1 + 1536996803884 p T + p^{21} T^{2} \)
23 \( 1 - 155899214954280 T + p^{21} T^{2} \)
29 \( 1 + 2400788707090758 T + p^{21} T^{2} \)
31 \( 1 - 2239820676947000 T + p^{21} T^{2} \)
37 \( 1 + 30785069383298890 T + p^{21} T^{2} \)
41 \( 1 - 103207571041281030 T + p^{21} T^{2} \)
43 \( 1 + 165557270617488124 T + p^{21} T^{2} \)
47 \( 1 - 66587216226477408 T + p^{21} T^{2} \)
53 \( 1 + 435422766592881630 T + p^{21} T^{2} \)
59 \( 1 + 5534365798259081316 T + p^{21} T^{2} \)
61 \( 1 + 7176205164722961202 T + p^{21} T^{2} \)
67 \( 1 + 15755449453068299812 T + p^{21} T^{2} \)
71 \( 1 + 26457854874259376232 T + p^{21} T^{2} \)
73 \( 1 - 13471249335464801450 T + p^{21} T^{2} \)
79 \( 1 + 16886125085525986840 T + p^{21} T^{2} \)
83 \( 1 - \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \)
89 \( 1 - \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \)
97 \( 1 - \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.44541243290921593737038989721, −14.44920245612142903557464689293, −13.19246029078025916833384099554, −11.96565407990803606952560186919, −10.73611087579463846148801264150, −7.63113534940665496266616984154, −6.03526718926988049921880519588, −4.71313088445062717712746740429, −3.35211043474268330135376328149, −1.76707195027819258339075285549, 1.76707195027819258339075285549, 3.35211043474268330135376328149, 4.71313088445062717712746740429, 6.03526718926988049921880519588, 7.63113534940665496266616984154, 10.73611087579463846148801264150, 11.96565407990803606952560186919, 13.19246029078025916833384099554, 14.44920245612142903557464689293, 15.44541243290921593737038989721

Graph of the $Z$-function along the critical line